Let \(C(G_n)\)  be the \(C^*\) -enveloping algebra of \(\mathbb {C}[G_n]\) . See [1]}, where the symbol \(C_u(G_n)\)  is used instead. Then the surjective \(*\) -homomorphism \(\mathbb {C}[G_{n+1}] \twoheadrightarrow \mathbb {C}[G_n]\)  extends to the \(C^*\) -level, i.e., \(r_n^{n+1} : C(G_{n+1}) \twoheadrightarrow C(G_n)\) .
Consider the \(\sigma \) -\(C^*\) -algebra
\(C(G) := \varprojlim C(G_n) = \Bigg \lbrace  (x_n) \in \prod _{n\ge 0} C(G_n)\,; r_n^{n+1}(x_{n+1}) = x_n \Bigg \rbrace ,\) 
